The geometric non-linear Schroedinger equation arises as in a simplification of the Landau-Lifschitz equations for a macroscopic ferro-magnetic continuum. The continuum is taken to be Rⁿ, where n = 1, 2, 3 and the equation is written

ĥ ĥt U = U ×DU

for U = (u₁,u₂,u₃) satisfying u²₁+u²₂+u²₃ = 1. For the domain R¹, this equation is equivalent to the integrable (focusing) non-linear Schroedinger equation. There is also an interpretation as the equation governing the tangent of a curve in R³ propagating in the direction of the binormal with speed equal to the curvature. The equation itself can be thought of as a non-linear Schroedinger equation governing the flow of a map from Rⁿ to S², and the target manifold S² can be replaced by any Kaehler manifold, yielding a family of non-linear Schroedinger equations whose behavior depends both on the dimension of the independent variable and on the geometry of the target. This equation belongs to the sequence of equations based on harmonic maps, including the harmonic map equation, the heat flow for the harmonic map equation and wave maps. We discuss the well-posedness of this equation, and some of the basic estimates which are used in proving long-time existence for non-linear Schroedinger equations.